Content: The first part of the module provides an introduction to vector calculus which is an essential toolkit for differential geometry and for mathematical modelling. After a brief review of line and surface integrals, div, grad and curl are introduced and followed by the two main results, namely, Gauss' Divergence Theorem and Stokes' Theorem. These theorems will be proved only in simple cases; complete proofs are best deferred until one has learned about manifolds and differential forms. The usefulness of these results in applications to flow problems and to the representation of vector fields with special properties by means of potentials will be emphasized. This leads to Laplace's and Poisson's equations which will be discussed briefly. The solution of these equations are discussed more fully in modules on partial differential equations. Cartesian coordinates are in many cases not well suited to a particular problem: for example, polar coordinates yield simpler equations for the flow of water in a cylindrical pipe. We will show how to represent div, grad and curl in general curvilinear coordinates, paying particular attention to spherical and cylindrical geometries.
The second part of the module introduces the rudiments of complex analysis leading up to the calculus of residues. The link with the first part of the module is achieved by considering a complex valued function of one complex variable as a vector field in the plane. Complex differentiability leads to the Cauchy-Riemann equations which are interpreted as conditions for the vector field to have both zero divergence and zero curl. Cauchy's theorem for complex differentiable functions is then established by means of the main integral theorems of vector calculus. Cauchy's integral formula which expresses the value of a complex differentiable function at a point as a line integral of the function on a contour surrounding the point is the key result from which the stunning properties of complex differentiable functions follow. Many real integrals can be computed using the so-called contour integration in the complex plane. Another interesting features is that complex functions can be expanded in so-called Laurent series around singular points in the plane. A Laurent series is a power series with eventually negative exponents.
Aims: This module aims to
- Teach a practical ability to work with functions of two or three variables and vector fields;
- Present the theorems of Gauss and Stokes as generalisations of the fundamental theorem of calculus to higher dimensions;
- Establish Cauchy's theorem in complex analysis as a consequence of the Cauchy-Riemann equations and the divergence theorems;
- Teach those rudiments of complex analysis which follow from Cauchy's theorem, namely, the Cauchy integral formula, Taylor expansions, Laurent series and residue calculus.
Objectives: On successful completion of this module, a student should
- Be able to calculate line, surface and volume integrals in general curvilinear coordinates;
- Be familiar with and use in a variety of contexts the fundamental results of vector calculus, namely, the divergence theorem and Stokes' theorem;
- Understand the relation between the existence of a scalar or vector potential of a vector field and the vanishing of the curl or divergence of that vector field and be able to calculate the potential when it exists,
- Be able to establish the Cauchy-Riemann equations for a complex differentiable function and establish Cauchy's theorem from the integral theorems of vector calculus;
- Be able to prove Cauchy's integral formula from Cauchy's theorem, and to use the integral formula to establish differentiability and series properties of complex differentiable functions;
- Be able to calculate Taylor expansions, residues and use them in the evaluation of definite integrals and summation of series.
There are a huge number of books that cover Vector and Complex Analysis at roughly the right level for this course. Comments on a selection of books that are useful for this module will be distributed at the first lecture. In addition lecture notes will be provided; see also